Exactness in SDP relaxations of QCQPs: Theory and applications

TL;DR

This paper explores the theoretical conditions under which SDP relaxations of QCQPs are exact, analyzing various notions of exactness and their implications for solving these NP-hard problems efficiently.

Contribution

It introduces and examines three notions of SDP exactness for QCQPs, providing new conditions and insights into when these relaxations yield exact solutions.

Findings

Objective value and convex hull exactness can be characterized geometrically.

The ROG property provides a sufficient condition for SDP exactness.

Various sufficient and necessary conditions for SDP exactness are discussed.

Abstract

Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems. In a QCQP, we are asked to minimize a (possibly nonconvex) quadratic function subject to a number of (possibly nonconvex) quadratic constraints. Such problems arise naturally in many areas of operations research, computer science, and engineering. Although QCQPs are NP-hard to solve in general, they admit a natural convex relaxation via the standard (Shor) semidefinite program (SDP) relaxation. In this tutorial, we will study the SDP relaxation for general QCQPs, present various exactness concepts related to this relaxation and discuss conditions guaranteeing such SDP exactness. In particular, we will define and examine three notions of SDP exactness: (i) objective value exactness -- the condition that the optimal value of the QCQP and the optimal value of its SDP relaxation coincide, (ii) convex hull exactness -- the condition that the convex hull of the QCQP epigraph coincides with the (projected) SDP epigraph, and (iii) the rank-one generated (ROG) property -- the condition that a particular conic subset of the positive semidefinite matrices related to a given QCQP is generated by its rank-one matrices. Our analysis for objective value exactness and convex hull exactness stems from a geometric treatment of the projected SDP relaxation and crucially considers how the objective function interacts with the constraints. The ROG property complements these results by offering a sufficient condition for both objective value exactness and convex hull exactness which is oblivious to the objective function. We will give a variety of sufficient conditions for these exactness conditions and discuss settings where these sufficient conditions are additionally necessary. Throughout, we will highlight implications of our results for a number of example applications.